Ideals in a multiplier algebra on the ball
HTML articles powered by AMS MathViewer
- by Raphaël Clouâtre and Kenneth R. Davidson PDF
- Trans. Amer. Math. Soc. 370 (2018), 1509-1527 Request permission
Abstract:
We study the ideals of the closure of the polynomial multipliers on the Drury-Arveson space. Structural results are obtained by investigating the relation between an ideal and its weak-$*$ closure, much in the spirit of the corresponding classical facts for the disc algebra. Zero sets for multipliers are also considered and are deeply intertwined with the structure of ideals. Our approach is primarily based on duality arguments.References
- Alexandru Aleman, Michael Hartz, John E. McCarthy, and Stefan Richter, Interpolating sequences in spaces with the complete Pick property, arXiv:1701.04885, 2017.
- William Arveson, Subalgebras of $C^*$-algebras. III. Multivariable operator theory, Acta Math. 181 (1998), no. 2, 159–228. MR 1668582, DOI 10.1007/BF02392585
- Errett Bishop, A general Rudin-Carleson theorem, Proc. Amer. Math. Soc. 13 (1962), 140–143. MR 133462, DOI 10.1090/S0002-9939-1962-0133462-4
- Lennart Carleson, Representations of continuous functions, Math. Z. 66 (1957), 447–451. MR 84035, DOI 10.1007/BF01186621
- Raphaël Clouâtre and Kenneth R. Davidson, Absolute continuity for commuting row contractions, J. Funct. Anal. 271 (2016), no. 3, 620–641. MR 3506960, DOI 10.1016/j.jfa.2016.04.012
- Raphaël Clouâtre and Kenneth R. Davidson, Duality, convexity and peak interpolation in the Drury-Arveson space, Adv. Math. 295 (2016), 90–149. MR 3488033, DOI 10.1016/j.aim.2016.02.035
- Brian Cole and R. Michael Range, $A$-measures on complex manifolds and some applications, J. Functional Analysis 11 (1972), 393–400. MR 0340646, DOI 10.1016/0022-1236(72)90061-4
- Kenneth R. Davidson, Michael Hartz, and Orr Moshe Shalit, Multipliers of embedded discs, Complex Anal. Oper. Theory 9 (2015), no. 2, 287–321. MR 3311940, DOI 10.1007/s11785-014-0360-8
- Kenneth R. Davidson and David R. Pitts, Invariant subspaces and hyper-reflexivity for free semigroup algebras, Proc. London Math. Soc. (3) 78 (1999), no. 2, 401–430. MR 1665248, DOI 10.1112/S002461159900180X
- Kenneth R. Davidson, Christopher Ramsey, and Orr Moshe Shalit, Operator algebras for analytic varieties, Trans. Amer. Math. Soc. 367 (2015), no. 2, 1121–1150. MR 3280039, DOI 10.1090/S0002-9947-2014-05888-1
- S. W. Drury, A generalization of von Neumann’s inequality to the complex ball, Proc. Amer. Math. Soc. 68 (1978), no. 3, 300–304. MR 480362, DOI 10.1090/S0002-9939-1978-0480362-8
- Theodore W. Gamelin, Uniform algebras, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1969. MR 0410387
- Devin C. V. Greene, Stefan Richter, and Carl Sundberg, The structure of inner multipliers on spaces with complete Nevanlinna-Pick kernels, J. Funct. Anal. 194 (2002), no. 2, 311–331. MR 1934606, DOI 10.1006/jfan.2002.3928
- Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. MR 0180696
- Håkan Hedenmalm, Closed ideals in the ball algebra, Bull. London Math. Soc. 21 (1989), no. 5, 469–474. MR 1005825, DOI 10.1112/blms/21.5.469
- G. M. Henkin, The Banach spaces of analytic functions in a ball and in a bicylinder are nonisomorphic, Funkcional. Anal. i Priložen. 2 (1968), no. 4, 82–91 (Russian). MR 0415288
- Kenneth Hoffman, Banach spaces of analytic functions, Prentice-Hall Series in Modern Analysis, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. MR 0133008
- Scott McCullough and Tavan T. Trent, Invariant subspaces and Nevanlinna-Pick kernels, J. Funct. Anal. 178 (2000), no. 1, 226–249. MR 1800795, DOI 10.1006/jfan.2000.3664
- V. Müller and F.-H. Vasilescu, Standard models for some commuting multioperators, Proc. Amer. Math. Soc. 117 (1993), no. 4, 979–989. MR 1112498, DOI 10.1090/S0002-9939-1993-1112498-0
- Peter Quiggin, For which reproducing kernel Hilbert spaces is Pick’s theorem true?, Integral Equations Operator Theory 16 (1993), no. 2, 244–266. MR 1205001, DOI 10.1007/BF01358955
- Walter Rudin, The closed ideals in an algebra of analytic functions, Canadian J. Math. 9 (1957), 426–434. MR 89254, DOI 10.4153/CJM-1957-050-0
- Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
- R. È. Val′skiĭ, Measures that are orthogonal to analytic functions in $C^{n}$, Dokl. Akad. Nauk SSSR 198 (1971), 502–505 (Russian). MR 0285899
- Hassler Whitney, Complex analytic varieties, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1972. MR 0387634
Additional Information
- Raphaël Clouâtre
- Affiliation: Department of Mathematics, University of Manitoba, 186 Dysart Road, Winnipeg, Manitoba, Canada R3T 2N2
- MR Author ID: 841119
- ORCID: 0000-0002-9691-2906
- Email: raphael.clouatre@umanitoba.ca
- Kenneth R. Davidson
- Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1
- MR Author ID: 55000
- ORCID: 0000-0002-5247-5548
- Email: krdavids@uwaterloo.ca
- Received by editor(s): April 18, 2016
- Published electronically: November 22, 2017
- Additional Notes: The first author was partially supported by an FQRNT postdoctoral fellowship and a start-up grant from the University of Manitoba.
The second author was partially supported by an NSERC grant. - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 1509-1527
- MSC (2010): Primary 46J20, 46E22
- DOI: https://doi.org/10.1090/tran/7007
- MathSciNet review: 3739183